I recently came across the concept of delta prime boundary conditions, which can be imposed on a function $\psi$ at the origin in one dimensional space (say, in the context of partial differential equations):
- $\frac{\partial\psi}{\partial x}$ is 'continuous' at $0$, in the sense that $\frac{\partial\psi}{\partial x}|_{0^+}=\frac{\partial\psi}{\partial x}|_{0^-}=:\psi'(0)$, meaning that both one sided limits exist and are equal (although $\psi$ doesn't have to be differentiable in the usual sense).
- The double sided limits of $\psi (0)$ exist and we have - $$\psi(0^+)-\psi(0^-)=-\sigma \psi'(0)$$
Where $\sigma\in\mathbb R$. Note that $\psi$ doesn't have to be continuous at $0$ (again - it need not be differentiable - only have one sided derivatives).
This gives me a certain boundary condition for any choice of $\sigma$. My question is - what happens to the condition as $\sigma\rightarrow \infty$? Is there a sense in which this limit can exist? What would be the corresponding boundary condition?
It seems to me that for me to take this limit (and if I want a bounded solution), I must have in the corresponding boundary condition that $\psi'(0)=0$. But what I'm interested in is - does $\psi$ now have to be continuous at $0$? We now have something of the form: $$\psi(0^+)-\psi(0^-)="-\infty\cdot 0"$$ Naively, we can't estimate the RHS and claim if $\psi$ is continuous at $0$. But maybe anyone here knows of a 'proper' way to take this limit (via distribution theory or something) so that we can know if maybe $\psi$ also needs to be continuous at $0$ for some reason?
This question is (partially on purpose) a bit vague and not too well defined - I have not stated which space of functions $\psi$ belongs to, etc. For me, this is part of the question - what properties would you require $\psi$ to fulfill for this question to 'make sense'? In what context can we say anything meaningful about this limit? There's probably more than one 'correct' answer to this - I'm interesting in anything you can suggest.
I'd be happy to hear all kinds of answers - intuitions, formal proofs (this is probably best), and even references to papers/books which do something similar to what I described. Keep in mind that I'm pretty new to this world of content, so ideally the more detailed the explanation - the better.
Thanks in advance!